Singlet magnetism in intermetallic UGa 2 unveiled by inelastic x-ray scattering

Using high resolution tender-x-ray resonant inelastic scattering and hard-x-ray non-resonant in-elastic scattering beyond the dipole limit we were able to detect electronic excitations in intermetallic UGa 2 that are highly atomic in nature. Analysis of the spectral lineshape reveals that the local 5 f 2 configuration characterizes the correlated nature of this ferromagnet. The orientation and directional dependence of the spectra indicate that the ground state is made of the Γ 1 singlet and/or Γ 6 doublet symmetry. With the ordered moment in the ab plane, we infer that the magnetism originates from the higher lying Γ 6 doublet being mixed with the Γ 1 singlet due to inter-site exchange, qualifying UGa 2 to be a true quantum magnet. The ability to observe atomic excitations is crucial to resolve the on-going debate about the degree of localization versus itineracy in U intermetallics.


I. INTRODUCTION
Actinide intermetallics show a wealth of fascinating phenomena that includes heavy-fermion behavior, hidden order or unconventional magnetism, unconventional superconductivity, the combination of ferromagnetism and superconductivity [1][2][3], orbital multicomponent [4], or spin-triplet superconductivity [5][6][7][8] with unusual topological properties [9].It is generally understood that those complex emergent properties originate from the intricate interplay of band formation and correlations involving the 5f electrons.It is, however, far from clear how to describe quantitatively the electronic structure of these systems, for example, whether an itinerant approach [2,3] or an embedded impurity model which includes explicitly the local degrees of freedom [10][11][12] would be the better starting point.The main problem is that many intermetallic uranium compounds, perhaps with the exception of UPd 3 [13,14], do not exhibit excitations in their inelastic neutron scattering data.It is therefore challenging to understand if remnants of atomic-like states are at all present in these compounds, let alone to pinpoint which multiplet and/or crystal-field state is actually occupied.
Here we investigate UGa 2 as a representative case for many metallic U compounds in which the relative importance between itinerancy and localization is at issue in explaining the physical properties.UGa 2 crystallizes in the * ASML Netherlands B.V., De Run 6501, 5504 DR, Veldhoven, The Netherlands hexagonal AlB 2 structure (space group P6/mmm) [15], with the U-U distances well above the Hill limit [16], and orders ferromagnetically below T c = 125 K with a small orthorhombic distortion [17].The moments are aligned in the ab-plane along the a crystallographic direction.The size of the uranium moment as determined by neutron diffraction [18] and magnetization [19][20][21] measurements amounts to about 3 µ B , quite a high value as compared to other magnetically ordered uranium intermetallics, and suggests a more localized nature of the 5f states in this binary.Inelastic neutron scattering, however, did not find crystal-field excitations; only magnons below 10 meV were observed [22].Attempts to explain the magnetism have been based on local f 2 and f 3 charge configurations [20,[23][24][25], and on approaches that include itinerancy [24,26].De Haas -van Alphen [20] and photoemission [27,28] experiments indicate that UGa 2 is neither localized nor itinerant.Spectroscopically, photoemission experiments are also not conclusive: core level data on bulk samples were interpreted as indicative for the localized nature of the 5f states, based on the satellite structure of the U 4f core level that looks very different from that of itinerant UB 2 [27,29].On the other hand, data on UGa 2 films [30] seem to support itinerancy, based on the fact that the satellites appear at different energy positions than in prototypical UPd 3 .
The observation of multiplets would provide direct evidence of the presence of atomic-like states.Furthermore, multiplets are a unique fingerprint of the configuration that determines the symmetry.Here resonant inelastic x-ray scattering (RIXS) is the ideal method because it covers a wide range of energy transfers.Already in   [34,35].However, the signal to background ratio of these f -f excitations is very small at the U Oedge because of the strong elastic tail in the extreme ultraviolet.A recent publication of soft RIXS data at the U N 4 edge (≈ 778 eV), also of UO 2 [36], encouraged some of the authors of this manuscript to repeat the N 4edge experiment with the same experimental set-up for the intermetallic large moment antiferromagnet UNi 2 Si 2 .
The result was discouraging, with absolutely no inelastic intensity observed [37].Another trial at the N 5 -edge (≈ 736 eV) of the hidden order compound URu 2 Si 2 gave the same negative result [38].Kvashnina et al., on the other hand, reported tender x-ray RIXS experiments at the U M 4 (≈ 3730 eV) and M 5 (≈ 3552 eV) edge with a resolution of 1 eV for UO 2 , and for the two intermetallic compounds UPd 3 and URu 2 Si 2 .Distinct excitations are observed at about 3 -7 eV (valence band into unoccupied 5f states) and 18-20 eV (U 5f to U 6p), both at the M 4 and M 5 edge [39].These data show that the realization of high-resolution tender RIXS at the U M -edges is the most promising direction to aim at, not only because of the expected stronger signal, but also because the tender x-ray regime does not require cleaving; it would even allow the confinement of samples.The latter would be a great advantage when performing experiments on U and especially actinide containing samples.
Here we utilize this new spectroscopic tool, namely high-resolution tender RIXS at the U M 5 edge to tackle the origin of the magnetism in UGa 2 .With tender RIXS, we were able to detect pronounced atomic multiplet states in the intra-valence band excitation spectrum of UGa 2 .We also present hard x-ray core-level non-resonant inelastic scattering data (NIXS, also known as x-ray Raman) in the beyond-dipole limit at the U O 4,5 -edge, confirming the RIXS result.Also in the high energy NIXS spectrum we observed states that are highly atomic in nature.Our analysis ultimately indicates that UGa 2 is a singlet ferromagnet.
Fig. 1 (b) depicts the experimental geometry where the scattering angle is set at 2Θ = 90 • to minimize the elastic intensity.The UGa 2 samples used for the experiments were grown with the Czochralski method [21] and their surface is the ab plane, i.e. it has the [001] orientation.
High resolution tender RIXS was performed at the Max-Planck RIXS end station (IRIXS) of the P01 beamline at Petra III/DESY in Hamburg.The instrument is unique, since it allows to perform RIXS experiments with tender x-rays (2.5 -3.5 keV) and good resolution [47].For example, a resolution of 100 meV can be achieved at the Ru L 3 -edge at 2840 eV.The IRIXS beamline uses the hard x-ray set-up [47].For the U M 5 edge at 3550 eV a diced quartz waver (112), pressed and glued on a concave Si lens has been used as analyzer crystal [48] B. NIXS with hard x-rays at U O4,5-edge NIXS with hard x-rays (10 keV) and large momentum transfer is dominated by higher-than-dipole transitions [49,50], which are more excitonic in contrast to the dipole contribution [51][52][53][54].The direction of the momentum transfer ⃗ q in NIXS plays an analogous role as the electric field vector ⃗ E in XAS and is sensitive to the symmetry of the crystal-field ground state.
The experiments are performed at the Max-Planck NIXS end stations of the P01 beamline at Petra III/DESY in Hamburg.A sketch and description of the NIXS experimental setup is shown in Fig. 2 of [55]. 10 keV photons are used.The average scattering angle 2Θ at which the analyzers are positioned is ≈ 150 o , thus yielding a momentum transfer of |⃗ q|=(9.6±0.1)Å-1 at elastic scattering.An instrumental energy resolution of about 0.8 eV FWHM is achieved.The sample is kept in a vacuum cryostat at T = 5 K.The O 4,5 edge of U is measured with momentum transfer ⃗ q parallel to the a and c crystallographic directions.

III. RESULTS
Fig. 1 (c) shows the experimental U M 5 -edge x-ray absorption (XAS) spectrum of UGa 2 .The large dots mark the photon energies used in this RIXS study, E res -4 eV (hν 1 ), E res (hν 2 ), and E res +4 eV (hν 3 ) with E res = 3552 eV.In Fig. 1 (d) the RIXS spectrum of UGa 2 is displayed for a wide energy range up to 8 eV energy transfer taken at the M 5 resonance (hν 2 ) with the sample angle of θ = 45 • .The spectrum exhibits sharp peaks below 2 eV that are on top of a broad feature that arises most likely from charge transfer excitations.The sharp peaks are very typical of local atomic-like excitations.
Fig. 2 shows a close-up of the first 2 eV of RIXS spectra that were measured with different incident energies, hν 1 , hν 2 and hν 3 .The data are normalized to the peak at 1.05 eV.The intensities of the peaks vary considerably with the incoming photon energy so that three inelastic excitations at 0.44, 0.70 and 1.05 eV can be identified.We assign these to intermultiplet f f transitions since the energies are too high for magnons and crystal-field excitations.
Full atomic multiplet calculations assuming a 5f 2 and alternatively a 5f 3 configuration were carried out to simulate the spectra.For this, the Quanty code [56] was used with the atomic values of the the spin-orbit constant and the 5f -5f and 3d-5f Slater integrals from the Atomic Structure Code by Robert D. Cowan [57] as input parameters, whereby the spin orbit constant was re- duced by 10% and the Slater integrals by 45% in order to take configuration interaction effects and covalence into account [58][59][60].These are typical reduction factors for uranium compounds [36].A crystal-field (CF) potential was always considered.CF parameters were taken from fits to the magnetic susceptibility or magnetization, for the 5f 2 from Ref. [25] and for the 5f 3 configuration from Ref. [20,23], or constructed to test different CF ground state wave functions.Furthermore, a Lorentzian broadening of about 6 eV in the intermediate state was used, based on the width of the M 5 XAS spectrum, and a Gaussian broadening of 150 meV to account for the experimental resolution.
Fig. 2 also shows the simulation for the 5f 2 and for the 5f 3 configuration.The calculation based on the 5f 2 reproduces the experimental data in terms of peak positions as well as variation of the peak intensities with incident energy.The vertical lines represent the histogram of the multiplet states and provide a straightforward labeling of the peaks.The 5f 3 simulation, on the other hand, does not reproduce the experimental data.It turns out that no matter how the reduction factors are tuned, an agreement cannot be achieved for 5f 3 (see Appendix VII A).Hence we conclude, the atomic-like states in UGa 2 are given by the 5f 2 configuration.
Next we determine the CF symmetry of the ground state.Here we ignore the slight orthorhombic distortion below T c [17] since it is only a very small magnetostriction correction to the hexagonal crystal-field analysis.In D 6h the hexagonal CF splits the nine-fold degenerate J = 4 Hund's rule ground state of the U 5f 2 configuration into three singlets and three doublets.These be written in the J z representation as: Although the CF splitting is below the resolution limit of the present RIXS experiment, it is possible to obtain information about the ground state symmetry by measuring the orientation dependence of the scattering signal [61].The two panels of Fig. 3(a) show the RIXS spectra for two incident energies (hν 1 and hν 3 ), whereupon each energy was measured for the three sample angles θ = 80 • , 45 • and 20 • .The θ rotation is in the [001]-[-210] plane, see Fig. 1(b).The intensities are again normalized to the peak at 1.05 eV energy transfer.A pronounced orientation dependence can be seen in the spectra for hν 1 , that has almost disappeared for hν 3 .
Again the data are compared to the full multiplet calculations.In Fig. 3(b) we start with the calculations using the 5f 2 crystal field parameters of Richter et al. [25].With this set of parameters, the ground state is the Γ 1 , the first excited state the Γ 6 at 3.3 meV, the second excited state the Γ 1  5 with sin ϕ = 0.81 at 5.9 meV, and all other states at 13 meV or higher.The calculations were performed for T = 35 K and include a molecular field I e of 1.78 meV as will be explained later.We observe that the calculations reproduce the experiment well: the simulation captures the strong orientation dependence for hν 1 in the correct sequence and its decrease for hν 3 .
To understand whether a different order of CF states would also be able to reproduce the experiment, we calculate the spectra for different CF ground states.To this end, we tuned slightly the CF parameters such that the desired CF state becomes the ground state and then we carried out the calculation for T = 0 K.The results are displayed in Fig. 3(c).We observe that a Γ 1 ground state, or a Γ 6 , or also a Γ 2  5 with cos ϕ ≈ 1 have the correct trend in the orientation dependence for hν 1 and its reduction at hν 3 .A Γ 3 , Γ 4 , or Γ 1  5 as ground state, on the other hand, produces an orientation dependence that is opposite to the experiment so that these three states can be excluded.The NIXS experiment below will show that also the Γ 2  5 cannot be the ground state.We thus conclude that the good simulation is based on the strong orientation dependence provided by the Γ 1 or the Γ 6 low lying states, which gets counteracted at 35 K by the Boltzmann occupation of a higher lying state with opposite orientation dependence, such as the Γ 1 5 .Fig. 4(a) shows the O 4,5 edge NIXS data of UGa 2 at 5 K for ⃗ q || c and ⃗ q || a, revealing a strong directional dependence.In Fig. 4(b) the (pseudo) isotropic U O 4,5 NIXS spectrum, constructed from I iso = (I q||c + 2I q||a )/3, is displayed and compared to atomic simulations without considering the crystal-field Hamiltonian.The Slater integrals for the 5f -5f and 5d-5f Coulomb interactions are reduced by about 40% with respect to their atomic values.The value of the momentum transfer in the simulation is set to |⃗ q| = 11.1 Å-1 in order to account for the U 5f radial wave function in the solid being different from the calculated atomic value.An arctangent type of background is added to account for the edge jump.A Gaussian broadening of 0.8 eV and a Lorentzian broadening of 1.3 eV account for instrumental resolution and lifetime effects, respectively.The simulations are performed both for an 5f 2 and 5f 3 configuration and also here only the f 2 simulation reproduces the experimental lineshape, whereas the f 3 does not (see Appendix VII B).This finding is fully consistent with the RIXS results.
Focussing now on the directional dependence, we cal- culate the spectra for each of the six possible crystal-field states.The results are displayed in Fig. 4(c).Comparison with experiment immediately excludes the Γ 3 and Γ 4 singlets, as well as the Γ 5 doublets for any range of the parameter ϕ (see equations above).For the Γ (1,2) 5 doublets only the extreme cases of ϕ = 0 o are shown, since the spectra for all other ϕ values fall between these two extremes.The Γ 1 singlet and the Γ 6 doublet, on the other hand, show the same directional dependence as the experiment, thus confirming the RIXS results.

IV. DISCUSSION
The above NIXS and RIXS results find that the 5f 2 configuration dominates the local electronic structure of UGa 2 and that the symmetry of the CF ground state is either given by the Γ 1 singlet and/or Γ 6 doublet.However, we can further exclude the Γ 6 doublet as ground state because it would yield an ordered moment along c and not in the ab-plane.Hence, the Γ 1 singlet state must be the lowest one in energy.Yet, the Γ 6 is also a necessary ingredient for the magnetism in UGa 2 as we will discuss in the following.
In a conventional local moment magnet the nonvanishing temperature independent moments are present at each lattice site and then order spontaneously at the transition temperature creating a self-consistent molecular field.This is basically a classical concept modified only by the influence of semiclassical quantum fluctuations which reduce the size of the ordered moment by a modest amount.A Γ 1 ground state would not carry a local moment so that the semiclassical picture of magnetic order does not apply, it rather must be classified as a true quantum magnet where the creation of the local moments and their ordering appears spontaneously at T c .This mechanism of induced magnetic order is caused by the non-diagonal mixing of Γ 1 with excited Γ 6 states due to the effective inter-site exchange coupling that forms the true ground state superposition below the ordering temperature.Induced quantum magnetism in singlet ground state systems has been explored in d 4 transition metal [62,63] or 4f 2 Pr materials (see Ref. [64] and references therein).In these cases the presence of multiplets is clear.Singlet magnetism is however rarely recognized in U compounds [64][65][66][67][68], where pinpointing the U 5f 2 configuration is already challenging (see also Appendix VII D).
Looking at the simple structure of CF states, we realize that indeed Γ 6 is the only possible excited state that has non-vanishing mixing matrix elements ⟨Γ 1 |J x |Γ 6 ⟩ for the in-plane total angular momentum operators (not for J z ) so that there can be no coupling to any other state when we restrict to the Hilbert space of the ground state multiplet (J = 4).This explains naturally that the ordered moment must lie in the hexagonal plane and at the same time the anisotropy of the paramagnetic susceptibility.For the induced moment mechanism of magnetic order to work, i.e. to produce a finite ordering temperature, the effective exchange I e must surpass a critical value.Here I e is the Fourier transform I(q) of the inter-site coupling I ij at the ordering vector q where I(q) is at its maximum.In a singlet ground state system with a Γ 1 -Γ 6 splitting energy ∆ a spontaneous induced moment can only appear when the control parameter ξ is larger than 1, with ξ = 2α 2 I e /∆ [64,66] and where σ is the degeneracy index of the Γ 6 states, with numerical value α = 3.1.The saturation moment at zero temperature then is given by m 0 /(g J µ B ) = ⟨J x ⟩ 0 = αξ −1 (ξ 2 − 1) 1/2 (g J = 0.8) which vanishes when approaching the critical value from above ξ → 1 + , and becomes equal to ⟨J x ⟩ 0 = α, that of a quasi-degenerate Γ 1 -Γ 6 system, when ξ ≫ 1, i.e.where the effective exchange strongly dominates over the splitting.In UGa 2 the moment of 3 µ B / U is close to the latter case of the exchange dominated regime.See also Appendix VII D.
The thermal occupation of higher levels has to be considered for the determination of the temperature dependence of ⟨J x ⟩ T and T c as a function of the exchange I e and CF splitting ∆.This can be done within our full multiplet calculation (including all angular momentum multiplets J and their respective CF multiplets) by solving iteratively the selfconsistency equation ⟨J x,y ⟩ T = n p n ⟨n|J x,y |n⟩ where E n (⟨J x,y ⟩ T ) and |n⟩(⟨J x,y ⟩ T ) are the eigenenergies and eigenstates in the presence of the molecular field I e ⟨J x,y ⟩ T for the given values of multiplet model parameters.Here p n = Z −1 exp(−E n /T ) with Z = m exp(−E m /T ) are the thermal level occupations.The saturation moment m 0 /µ B = g J ⟨J x ⟩ 0 may then be plotted as function of the splitting ∆ and exchange I e as shown in Fig. 5.Here the CF parameters from Ref. [25] (apart from the off-diagonal A 6  6 ) are scaled to modify the splitting ∆.It would be interesting to measure ∆ using Raman spectroscopy [69].For ∆ = 3.3 meV and I e = 1.78 meV, corresponding to m 0 (∆, I e ) ≈ 3 µ B (one point on the white line in Fig. 5), we obtain T c ≈ 125 K, in agreement with experiment.For completeness, we finally examine the impact of the self consistent molecular field with I e = 1.78 meV on the RIXS (see Fig. 3 (b)) and NIXS (see Fig. 4 (a)) spectra.Here the CEF scheme giving ∆ = 3.3 meV and the Boltzmann population of excited states is also considered.We see that the molecular field has little impact on the spectra, since, although mixed, the Γ 1 and Γ 6 show very similar lineshapes by themselves.

V. CONCLUSION
In summary, with tender RIXS at the U M 5 edge and hard x-ray NIXS at the U O 4,5 -edge we have unveiled the U 5f 2 multiplets inUGa 2 and shown that the magnetism is determined by the U 5f 2 configuration with a Γ 1 singlet ground state and a Γ 6 doublet nearby.UGa 2 , therefore, classifies as a quantum magnet.The origin of the induced magnetic order is due to the non-diagonal mixing of Γ 1 with excited Γ 6 states due to the effective inter-site exchange coupling below T c .Fig. 6 shows simulated f 3 RIXS spectra with 5f − 5f Slater integral reduced of 45%, 40% and 75%.The spectra are calculated with incident energies hν 1 , hν 2 and hν 3 , and with the crystal-field parameters from [23].

C. Crystal-field parameters and ground state symmetry
Table VII C summarizes the crystal-field parameters and the corresponding ground state symmetries used in the RIXS calculations.For the Γ 5 states the relative J z = | ± 4⟩ and J z = | ∓ 2⟩ contribution is specified.The parameters giving a Γ 1 ground state are taken rom Ref. [25].

D. Basics of induced moment magnetism
In this work the magnetism of UGa 2 is interpreted in terms of a localized model consisting of 5f CEF states for J = 4.The on-site exchange interaction (resulting from Anderson-type on-site hybridisation and Coulomb repulsion) between conduction and f-electron is assumed to have been eliminated leading to an effective RKKYinteraction I ij between 5f states on different sites i, j.I e is the Fourier transform I(q) of the inter-site coupling I ij at the ordering vector q where I(q) is at its maximum.Restricting to FM with q = 0 and to nearest neighbor terms only, the effective interaction is given by I e = zI nn (z=coordination number).If the 5f ground state were degenerate and carried an effective moment (i.e.having nonzero matrix elements of J within the multiplet) a quasiclassical ferromagnetic order would appear for any size of I e where moments are simply aligned at a temperature T C ∼ I e .Here, however, the lowest 5f states are nonmagnetic singlet Γ 1 ground state and Γ 6 doublet excited state at energy ∆.Due to their absent moments the FM order in UGa 2 can only appear through a more subtle mechanism called 'induced order'.This mechanism is well established for several 4f Pr and 5f U compounds with nonmagnetic low lying CEF states as in the present case.We refer to previous Refs.[64,66,[70][71][72][73][74] for the detailed discussion of the subject.Although the Γ 1 , Γ 6 states do not carry a moment there are nondiagonal matrix elements α/ √ 2 = ⟨Γ 1 |J x |Γ 6σ ⟩ (σ = 1, 2) of in-plane dipolar moment J x (and similar for J y ) connecting them accross the CEF gap ∆.This means that n.n.inter-site interaction terms like I ij J x (i)J x (j) are able to mix the excited state Γ 6 into the noninteracting ground state Γ 1 and form spontaneously a new magnetic ground state at each site which is a superposition |Γ ′ 1 ⟩ = u|Γ 1 ⟩ + v|Γ 6 ⟩ (and similar for the excited state).In this way the ground state moment appearance and its ordering happens simultaneously.The size of the ordered moment is then ⟨J x ⟩ = 2uvα(n ′ 1 − n ′ 6 ) where n ′ 1,6 denote the thermal occupations of the CEF states which also depend on ⟨J x ⟩.This represents a molecular field equation for the induced moment ⟨J x ⟩.When temperature is lowered the occupation difference increases which may lead to a nonzero induced moment, provided the prefactor in the above equation is sufficiently large.This can be evaluated as a condition for the control parameter ξ = 2α 2 I e /∆ > 1 to achieve a finite T C and a saturation moment at T = 0 given by ⟨J x ⟩ 0 = αξ −1 (ξ 2 − 1) 1 2 .At zero temperature varying ξ across the quantum critical point (QCP) ξ = 1 we obtain a quantum phase transition from the paramagnetic (ξ < 1) to the (ferro-) magnetic (ξ > 1) state.In particular close to the QCP the induced moment quantum magnetism shows anomalous dependence of small saturation moment and low ordering temperature on the control parameter and is quite different from the quasiclassical magnetism where the influence of quantum fluctuations on moment and transition temperature is moderate.

Fig. 1
Photon energy (eV) . The instrument 150 meV Gaussian response function at the U M 5 edge is estimated by measuring a carbon tape.The experiment was performed with horizontal polar-ization of the the incident photons, a scattering angle 2Θ = 90 o to minimize elastic intensity and sample angles of θ = 20 • , 45 • and 80 • (see Fig. 1(b)).Temperature was kept at 35 K.

FIG. 3 .
Fig.3 FIG. 4. (a) NIXS spectra for ⃗ q || a and ⃗ q|| c together with the simulation using the crystal field parameters of Richter et al. [25] and a molecular field Ie of 1.78 meV.(b) (Pseudo)isotropic data and full multiplet simulations without crystal-field for the U f 2 and U f 3 configurations.(d) Simulated NIXS spectra for each of the six crystal-field states of the f 2 together with the corresponding charge densities.

FIG. 5 .
FIG. 5. Magnetic moment m0 at zero temperature as a function of the Γ1 − Γ6 splitting ∆ and the effective exchange Ie.The white line denotes the contour with m0 = 3 µB